How do you find the exact value of #cos(u-v)# given that #sinu=5/13# and #cosv=-3/5#?

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marfre Share
Mar 20, 2017

Answer:

#cos(u-v) = -16/65#

Explanation:

Find #cos u# using Pythagorean Theorem to find the adjacent side. Assume angle #u# is in the first quadrant:

  1. #13^2 = 5^2 + a^2#
  2. #a^2 = 169 - 25 = 144#
  3. #a = 12#
    So #cos u = 12/13#

Find #sin v# using Pythagorean Theorem to find the adjacent side. Assume angle #v# is in the second quadrant:

  1. #(-3)^2 + b^2 = 5^2#
  2. #b^2 = 25 - 9 = 16#
  3. #b = 4#
    So #sin v = 4/5#

Use the difference formula #cos(u-v) = cos u cos v + sin u sin v#:
#cos(u-v) = 12/13*-3/5 + 5/13*4/5 = -36/65 + 20/65 = -16/65#

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